g_KM = c/6 for the compression direction of a general conformal net (conjecture; only the analytically continued value exists)

Status: conjectural — believed, with partial evidence; not proved.

Area: Kubo-Mori (relative entropy) coefficient s2 · Card type: conjecture · Knowledge-base id: const-gkm-general-nets

Statement

Conjecture: for a general diffeomorphism-covariant net of central charge c, the Kubo-Mori form of the zero-collar compression direction is g_KM = c/6, i.e. the free-fermion value r/6 (card const-gb) scaled by c/r. Status: Note 7 obtains c/6 only through the formal modular-frame prescription with the analytically continued g-hat (universality_normalization.tex:301-302, 'used here formally, as a computational prescription'; eq:result-general); no admissible representative of the tangent vector is affiliated with A(I), and for fields affiliated with an interval algebra A(K), K containing I, the Kubo-Mori form is a local Sobolev functional whose infimum over admissible gauges is (11 + 5 sqrt 5) c/12 = 11.09 x c/6 (rigor/kubo_mori_gauge_lemma.tex eq:mainvalue); the Kubo-Mori second variation for generators not affiliated with the interval algebra is open (Gap QK2 of rigor/kubo_mori_second_variation.tex; card km-exact-identity). Related numerics support s2 = c/12 for the relative entropy along the recovery curve (free fermion 0.0837(5), free boson 0.083(2); card conj-s2), where the bracket 0.034 c <= liminf s^{-2} D <= limsup s^{-2} D <= 0.924 c is conditional at both ends (the upper bound under the uniform-integrability hypothesis (V), which is open, Gap RK1; the lower bound modulo Araki's perturbation expansion, whose sources were not obtained, Gap RK2; rigor/kubo_mori_gauge_lemma.tex, status table; called rigorous here until 2026-09-24).

How to verify

universality_normalization.tex l. 296-302 (formal prescription, gauge infimum, bracket) and eq:result-general; rigor/kubo_mori_gauge_lemma.tex eq:mainvalue and Sec. sec:gauge; rigor/kubo_mori_second_variation.tex Gap QK2 (l. 822).

Evidence

• universality_normalization.tex — document, at eq:result-general
• rigor/kubo_mori_gauge_lemma.tex — document, at eq:mainvalue
• rigor/kubo_mori_gauge_lemma.tex — document, at sec:gauge
• rigor/kubo_mori_second_variation.tex — document, at sec:open

Related claims

• Free-fermion one-particle metrics of the compression direction: g_B = 2r/(3 pi^2), g_KM = r/6 (c = r) — proved
• s2 = c/12 for general nets — conjectural
• Exact relative-entropy identity on unitary orbits (bounded generators; affiliated generators with Omega in D(A)) — proved
• km-gauge-lemma-refuted — refuted, no page of its own

Review

Review record: passed 2026-09-24 by REF-CARD-OVER.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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